Why are the faces skipped?
The first gif shows that you have faces which aren't properly connected to the surrounding vertices. This can be seen because the face center is marked with a little black square.



The top face is an ngon that consists of eight vertices (marked in blue) and the center vertex isn't part of it.



In order to fix this, switch to face select mode (either by clicking on the icon or using the 3 shortcut in Blender 2.8x), select the top face and delete it (X > Faces). 





Switch back to vertex select mode and select four vertices that are supposed to form a face. Press F to create a face and repeat this step with the next four vertices until you have created all four faces.




The two faces on the side are ngons as well and consist of six vertices each. The vertex you're moving in the second gif is part of both faces, but the two horizontal edges aren't.

Switch to face select mode and select both problematic faces. Then press X > Faces Only, which will remove the faces but keep the vertical edge.



Once the side faces are deleted, switch to vertex select mode, select four vertices and create a face with F. Repeat this step until you've created all four faces.